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Langmuir 1996, 12, 5022-5027
Spontaneous Curvature-Induced Rayleigh-like Instability in Swollen Cylindrical Micelles R. Granek Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel Received April 22, 1996. In Final Form: July 5, 1996X We study a Rayleigh-like instability of an oil in water cylindrical “droplet” in the presence of excess surfactant. The surfactant monolayer is modeled by an interfacial bending energy with a nonvanishing spontaneous curvature C0. We find that, even when surfactant reduces the interfacial tension to a vanishingly small value, the bending energy with spontaneous curvature can destabilize the cylinder if its radius R is greater than C0-1. The cylinder deforms into “beads” which can eventually break up into spherical droplets whose radius we predict. Unlike in the classical Rayleigh case, we find that the total interfacial area is usually increasing. Using our theory, we discuss the kinetics of the transition from a microemulsion phase of swollen cylindrical micelles to a droplet phase, following a sudden increase in the spontaneous curvature parameter.
I. Introduction The instability of cylindrical fluid objects against peristaltic deformations and subsequent breakup into drops, the well-known Rayleigh instability, is involved in many daily experiences, e.g., the instability of jets. More recently, the interest in the Rayleigh instability has been revived in conjunction with the action of optical tweezers on cylindrical bilayer vesicles.1-3 It is argued that the tweezers induce a tension in the membrane, which is sufficiently large to overcome the stabilizing bending forces. On the practical side, the Rayleigh instability is sometimes used in industry for dispersing one fluid in another.4 It might also be relevant for enhanced oil recovery from porous rocks, in cases where the pore geometry is roughly cylindrical and when water is better at wetting the rock than oil is. An important question associated with (say) oil in water (or vice versa) tubes is the effect of surfactant. The most dramatic effect of surfactant adsorption is to reduce the interfacial tension.4 At the same time, the bending energy becomes increasingly important as the interfacial surfactant concentration rises up.5 In the absence of spontaneous curvature, the latter stabilizes the cylinder against small deformations.1 One might have expected that the effect of surfactant adsorption would be to slow down the Rayleigh instability, and even to completely inhibit it. However, a single surfactant monolayer is usually characterized also by its spontaneous curvature.5 In practice the spontaneous radius of curvature ranges between a few angstroms to ∼103 Å. If the cylinder radius is different from this radius, one may consider the possibility that the cylinder will be unstable against its breakup into droplets whose radius better matches the spontaneous radius. To examine this possibility, consider the following argument, in which we disregard the X Abstract published in Advance ACS Abstracts, September 15, 1996.
(1) Bar-Ziv, R.; Moses, E. Phys. Rev. Lett. 1994, 73, 1392. (2) Granek, R.; Olami, Z. J. Phys. II 1995, 5, 1349. (3) Nelson, P.; Powers, T.; Seifert, U. Phys. Rev. Lett. 1995, 74, 3384. (4) Miller, C. A.; Neogi, P. Interfacial PhenomenasEquilibrium and Dynamic Effects; Surfactant Science Series Vol. 17; Marcel Dekker: New York, 1985. (5) Menes, R.; Safran, S. A.; Strey, R. Phys. Rev. Lett. 1995, 74, 3399. Safran, S. A. Statistical Thermodynamics of Surfaces, Interfaces, and Membranes; Frontiers in PhysicssVol. 90; Addison-Wesley: New York, 1994; pp 248-252. Safran, S. A.; Turkevich, L. A.; Pincus, P. J. Phys. Lett. 1984, 45, L-69.
S0743-7463(96)00383-6 CCC: $12.00
question of whether the cylinder is truly unstable or just metastable (discussed in section II) and simply compare the energy of small separate droplets with the energy of a cylinder of radius Rc > C0-1. Assuming that the interfacial tension σ vanishes, and neglecting for simplicity the Gaussian rigidity, the minimal droplet energy is obtained for the droplet radius RsC0 ) 2, in which case it vanishes (and so is lower than the cylindrical energy). Conservation of volume then dictates the number of droplets to be formed. Moreover, taking, say Rc ) 1.1 × C0-1, we find that droplets of radius in the range 1.82 < RsC0 < 2.24 also have a lower energy than that of the cylinder. It thus remains to explore whether the cylinder is truly unstable against small deformations or it is only metastable (section II). On the experimental side, we believe that the instability we describe here is most relevant for the transition from a microemulsion phase of swollen cylindrical (wormlike or rodlike) micelles to a droplet (swollen spherical micelles) phase.5 This transition is often observed when the micelles are swollen above a certain degree, e.g., for water to surfactant ratios w above 5 in certain reverse micelles of transition-metal/Aerosol-OT complexes in alkane solvents or for w above 7 for reverse micelles of DDAB (didodecyldimethylammonium bromide) in cyclohexane.6 A similar situation exists for reverse wormlike micelles of Soybean lecithin in organic solvents,7 with a swelling limit depending on the organic solvent (5 < w < 10). (In systems of direct-wormlike micelles of ionic surfactants, e.g., CTAB or CTAC in brine solutions,8,9 this transition presumably occurs at a vanishingly small swelling limit, although no systematic studies of this particular question have so far been reported.) Assuming that a system of swollen cylindrical micelles exists, the scenario we depict in this paper can thus be used to describe the kinetics of this transition upon a sudden change in the interfacial free energy parameters, e.g., in the spontaneous curvature (section III). The latter could be achieved, e.g., by a sudden change in temperature orsin the case of ionic surfactantssin salt concentration. We note however that we shall describe only the initial (but dominant) stages (6) Eastoe, J. Langmuir 1992, 8, 1503 and references therein. (7) Schurtenberger, P.; Scartazzini, R.; Magid, L. J.; Leser, M. E.; Luisi, P. L. J. Phys. Chem. 1990, 94, 3695. Schurtenberger, P.; Cavaco, C. Langmuir 1994, 10, 100. (8) Kern, F.; Zana, R.; Candau, S. J. Langmuir 1991, 7, 1344. (9) Porte, G. In Micelles, Membranes, Microemulsions and Monolayers; Gelbart, W. M., Ben-Shaul, A., Roux, D., Eds.; Springer: New York, 1994.
© 1996 American Chemical Society
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Langmuir, Vol. 12, No. 21, 1996 5023
of the kinetics. Slower processes, involving the expulsion of the excess (say) oil due to the increase in total interfacial area, will not be described here. II. Spontaneous Curvature Instability II.a. Bending Free Energy in Cylindrical Coordinates. We consider a cylindrical thread of oil in water surfactant solution. It is assumed that the surfactant adsorbs strongly to the interface. As a result, the interfacial tension is expected to be vanishingly small and so the bending free energy becomes important. The latter may be described by the Helfrich free energy10
∫
1 Fκ ) κ ds (H - C0)2 2
(1)
where
geometries and equate it with eq 3. We thus find that the (saturation) interfacial tension σ for the underlying cylinder is σ = 0 for this case. If the column is injected into a (dilute) bilayer lamellar phase, similar considerations show that σ ) -1/2κC02. A more practical situation (discussed in section III) is the case of a sudden change in the spontaneous curvature C0 in a microemulsion phase of cylindrical “droplets” (swollen rod- or wormlike micelles). In this case the source of surfactant to an unstable tether is some other (initially identical) tethers. The energy difference per unit (actual) area ∆ between a deformed and undeformed cylinder is
(
)
1 1 1 ∆ ) κ(H - C0)2 - κ - C0 2 2 Rc
2
(5)
and so σ for the cylinder under study is
H)
1 1 + R1 R2
is the curvature (R1 and R2 are local radii of curvature), C0 is the spontaneous curvature (assumed positive, for simplicity), and κ is the bending modulus. (Since the initial deformations do not involve changes of topology, we may drop the Gaussian curvature, which is a topological invariant. It nonetheless can affect the final stage of droplet breakoff, which we avoid discussing explicitly.) If the interfacial tension σ does not vanish, the total free energy is
F ) Fκ + Fσ
(3)
∫ds σ
(4)
Fσ )
The value of σ depends on the actual surfactant density at the interface and is, in principle, a dynamic variable. It can affect the stability analysis, since a deformed cylinder must usually change its area. In order to formulate the problem uniquely, we must therefore specify the value of σ. Its saturation value depends on the “cmc” (critical micelle concentration) value of the chemical potential in the phase which is put in contact with the interface. Here we mean “cmc” in the broad sense, i.e., the point at which an increase of surfactant concentration does not yield an increase of the chemical potential. It is useful to express this tension in terms of the bending energy parameters κ and C0. This can be done to some approximation when the aggregates (or droplets) in the contact phase have a well defined geometry. The approach is to consider the overall free energy change in the system when the area of the interface under consideration is increased by an area element ds, which has just the same surfactant concentration as the rest of the interface. This means that surfactant has to be transferred from the bulk to the interface. The calculated free energy change is equated with eq 3 (which we subsequently use to describe only the cylindrical interface, no matter what is the source of surfactant), thereby determining the value of σ. As an example, consider an oil column which is injected into a (spherical) droplet microemulsion phase. The droplet radius is thus expected to minimize the bending energy, implying a radius Rs = 2/C0. The surfactant adsorbed on the droplets is now treated as a reservoir for the (possibly deformed) cylindrical interface. Assuming that the area per molecule is roughly the same for both spherical and cylindrical geometries, we calculate the energy difference per unit area ∆ between the two (10) Helfrich, W. F. Z. Naturforsch 1973, 28c, 693.
(
)
1 1 σ)- κ - C0 2 Rc
(2)
2
(6)
Thus σ depends on both Rc and C0. Before the C0 jump is made, we expect that Rc minimizes the bending energy, Rc = C0-1, and so σ = 0. According to eq 6 a jump of C0 changes also σ. For a large jump C0 . 1/Rc, σ = -1/2κC02. For an arbitrary jump -1/2κC02 < σ < 0. II.b. Stability Analysis. We now consider small axisymmetric deformations of the emulsion (or microemulsion) “vesicle” from its cylindrical shape, conserving the total enclosed volume11 and taking the initial radius Rc to be much smaller than L. We also keep constant the tube length L,11 since a change of L would require a Poiseuille-type flow along the tube, which is extremely slow relative to the evolution of short wavelength deformations. The deformation is described by the shape function u(z) (∫L0 dz u(z) ) 0) by writing the local tube radius as R + u(z), where the average radius R (*Rc) is yet to be determined. Conservation of volume leads to the relation between R and Rc
Rc2L ) R2L +
∫0Lu(z)2 dz
(7)
which, in Fourier space, becomes
R2 ) Rc2 -
∑q UqU-q
(8)
The conservation of the volume and length of the cylinder leads to a change of the area. For small deformations, |∂u/∂z| , 1, the area is
∑q q2UqU-q
S = 2πRL + πRL
(9)
or, using eq 8,
S S0
=1+
1
∑q [(qRc)2 - 1]UqU-q
2Rc
2
(10)
where S0 ) 2πRcL is the area of the undeformed cylinder. We proceed to calculate the free-energy in the harmonic approximation (i.e., to order u2), including both bending free-energy and surface tension. This is done in a similar way to that described in refs 2 and 12, and we shall omit the details here. The free-energy difference between a (11) Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability; Dover: New York, 1981; Chapter XII. (12) Zhong-can, O.-Y.; Helfrich, W. Phys. Rev. 1989, A39, 5280.
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Granek
deformed and undeformed cylinder, ∆F, is found to be
∆F =
πκL Rc3
∑q γ(q)UqU-q
(11)
The case σ˜ < (1 + 4C ˜ 0)/2, however, is strongly influenced by the spontaneous curvature. Consider first C ˜ 0 < 1. From the condition γ(q˜ m) < 0 we find that the cylinder is unstable for small axisymmetric deformations in the following two ˜ 0)/2 and σ˜ < σ˜ -, where tension regimes, σ˜ + < σ˜ < (1 + 4C
1 σ˜ ( ) (4C ˜ 0) ˜ 0 - 3 ( 4x2x1 - C 2
where
γ(q) )
1 3 - σ˜ + σ˜ - - 2C ˜ 0 q˜ 2 + q˜ 4 2 2
(
)
(12)
and where we have defined
q˜ ) qRc
(13)
C ˜ 0 ) C0Rc
(14)
˜ 02 σ˜ ) σRc2/κ + 1/2C
(15)
Importantly, the spontaneous curvature enters already in the harmonic approximation. This contrasts with the small deformations of a flat surface, in which the spontaneous curvature enters only in higher order odd terms which break the symmetry between up and down. It is a consequence of using a cylindrical geometry, in which the symmetry between inside and outside is already broken. (A similar situation, but without coupling of the spontaneous curvature to the wavelength, occurs in deformations of a spherical droplet.12-14) Equation 11 is similar to the result of Zhong-can and Helfrich for the energy of a deformed cylindrical vesicle.12 It differs from their (final) result because of the different ensemble used.15 In the study of Zhong-can and Helfrich the interior volume is allowed to vary accompanying a suitable free-energy penalty associated with work done against the Laplace pressure. This is a reasonable choice for membrane vesicles which can exchange the solvent between interior and exterior on the experimental time scale. Here, on the other hand, we consider a single microemulsion “vesicle” whose interior fluid does not mix with the exterior one and so cannot exchange with it, nor can it exchange with a reservoir. Thus, for an incompressible fluid, the enclosed volume must be conserved. According to eq 11, if γ(q) is negative in a certain range of q, the cylinder will be unstable against deformations in this band. Consequently, there is a fastest growing mode q*. While this mode has to be determined from dynamic analysis,11 it is strongly influenced by the wavenumber qm that minimizes γ(q). We shall therefore discuss this wavenumber, bearing in mind that it is not exactly q*. In addition, discussing this wavenumber is useful for the stability analysis, since an unstable band exists if γ(qm) < 0. Minimizing γ(q) over q, we thus find the energetically most unstable mode
˜ 0 - 2σ˜ )1/2/2 q˜ m ) (1 + 4C
(16)
Equation 16 is applicable so long as σ˜ < (1 + 4C ˜ 0)/2. If the opposite is true, σ˜ > (1 + 4C ˜ 0)/2, the physically accepted minimum is at q˜ m ) 0, which is similar to the classical Rayleigh case. In the latter situation, the spontaneous curvature has little effect and the tube is unstable (γ(q)0) < 0) for tensions σ˜ > 3/2, as further discussed in refs 1-3 for the case C0 ) 0. (13) Milner, S. T.; Safran, S. A. Phys. Rev. 1987, A36, 4371. (14) Peterson, M. A. J. Appl. Phys. 1985, 57, 1739. (15) Equations 11 and 12 are nevertheless consistent with some intermediate results of Zhong-can and Helfrich,12 provided that volume conservation is accounted for.
(17)
The two branches are separated by a gap which shrinks ˜ 0 ) 1, i.e., when the cylinder to the point σ˜ ( ) 1/2 when C radius is equal to the spontaneous radius of curvature. ˜ 0)/2, which Note that when C ˜ 0 < 1/2, we get σ˜ + > (1 + 4C means that the upper branch is not relevant; the tube is unstable only for σ˜ < σ˜ - (with q˜ m given by eq 16) and for σ˜ > 3/2 (with q˜ m ) 0). On the other hand, for C ˜ 0 > 1 the cylinder is unstable for all surface tensions σ˜ . As σ˜ passes through (1 + 4C ˜ 0)/2 from below, q˜ m changes from the finite value given in eq 16 to q˜ m ) 0. We see that varying the two effective parameters σ˜ and C ˜ 0 opens up a rich variety of different scenarios. As an explicit example consider the special value σ˜ ) 1/2C ˜ 02, i.e., σ ) 0. As discussed in section II.a, this value of σ corresponds to a surfactant saturated interface of a (nonequilibrated) oil column injected into a droplet microemulsion phase. For this case fluctuations ˜ 0 > 1 there at all wavelengths are stable if C ˜ 0 e 1.16 For C is an unstable band, q˜ - < q˜ < q+, where q˜ ( are the positive roots of γ(q˜ ) ) 0. The energetically most unstable mode is (c.f., eq 16)
˜0 - C ˜ 20)1/2/2 q˜ m ) (1 + 4C
(18)
It is instructive to divide the unstable regime, C ˜ 0 > 1, into two subregimes: (i) 1 < C ˜ 0 < 3, for which q˜ m > 1. As seen from eq 9, in which the sum is dominated by q˜ m, this implies that the total area is increasing by the deformation. (ii) C ˜ 0 > 3, for which q˜ m < 1 and the total area is decreasing. This case is dominated by the curvature independent constant 1/2κC02, that contributes to the surface tension. Case ii is thus more similar to the classical Rayleigh instability; however, unlike the Rayleigh case, for 3 < C ˜0 < 2(1 + x5) the value of q˜ m is finite. For C ˜ 0 > 2(1 + x5) we see that qm vanishes or, more precisely, obtains its low cutoff value qm ) π/L, as in the Rayleigh case. In the latter situation, however, the fastest growing wavenumber q* is finite.2,17 (In fact, the dynamics also influences somewhat the regime 3 < C ˜ 0 < 2(1 + x5) and at the low end of this regime we find q˜ * > 1; see section III.) In general, by increasing σ the problem resembles more and more the classical Rayleigh instability. In practice, ˜ 0) are usually not however, σ and C0 (and so σ˜ and C independent variables, as discussed in section II.a. Changing physical parameters, such as temperature or salt content, will usually change both σ and C0 simultaneously. The explicit relation between the two parameters does depend on the specific problem considered. For the problem of an oil column injected into a droplet microemulsion phase the initial value of σ˜ is 1/2C ˜ 02. On the other hand, for a C0 jump experiment in a swollen cylindrical micelle phase, we have σ˜ ) -1/2 + C ˜ 0. This more practical situation is studied next. III. C0 Jump in Swollen Cylindrical Micelles As already discussed, we consider an equilibrium microemulsion phase of long tether-like drops. In more (16) We consider only positive C0. (17) Tomotika, S. Proc. R. Soc. London 1932, A150, 322.
Instability in Swollen Cylindrical Micelles
Langmuir, Vol. 12, No. 21, 1996 5025
familiar terms, these are simply swollen cylindrical micelles. They can have either rodlike or wormlike shapes and can be in either dilute or semidilute regimes (the definitions are analogous to those for polymer solutions). The stability of these micelles can be understood by their ability to match the cylindrical radius with the spontaneous radius of curvature. This occurs in intermediate surfactant concentrations which are too low for creation of spherical droplets and for small (in magnitude) Gaussian rigidities, |κj|/κ , 1 (κj < 0), which otherwise also favor spherical droplets.5 It is well established experimentally that the micelles can be swollen (e.g., by water, in the case of reversed micelles) only up to a certain degree, above which one observes a transition to a spherical droplet phase.6 This transition has so far been studied, either theoretically5 or experimentally,6,7 only in equilibrium. Here we would like to study the initial dynamics of this transition following a sudden jump in one of the relevant variables. From a practical viewpoint this cannot be the volume fraction of the swelling fluid, as in some equilibrium studies.6 One can however induce a jump in the parameter C0. This can be achieved, e.g., in a temperature jump (T jump) experiment,18,19 although the complicated dependence on the temperature of the other parameters, such as the bending rigidity κ, can make it hard to interpret the results. A more controlled experiment in the case of ionic surfactant could be a fast precipitation of salt. This however will be limited by the finite time it will take the ions to diffuse and equilibrate in the system. Disregarding these experimental difficulties, we now consider a perfect C0 jump (increase) experiment. For clarity we shall assume that the solvent is water and the micelles are swollen by oil. As shown below, the (initial, nonequilibrium) droplet phase which will be formed has a larger interfacial area. This is also expected from an equilibrium viewpoint, since the droplet radius should match the spontaneous radius of curvature. Since the total amount of surfactant in the system is fixed, and since most of it lies at the interface, some excess oil has to be expelled out of the newly formed phase (emulsification failure). This implies that by some stochastic evolution some of the tethers will give away their surfactant in favor of other unstable tethers, followed by a slow expulsion of their (oil) content. These aspects are too complicated to be considered here. We shall focus on one tether which is unstable, assuming that the nearest neighboring tethers remain stable, at least during the time it takes for the tube to complete its breakup into droplets. In the discussion in section II.a we concluded that the effective tension to be used for this problem is
σ˜ ) -1/2 + C ˜0
(19)
same is true for the fastest growing mode q˜ *.20 This implies that the area will increase in this deformation, as expected. We now wish to find the time scale for droplet breakoff and the size of droplets, estimated from the wavelength of the fastest growing mode. For this purpose it might not be entirely sufficient to assume that the interfacial tension, bending constant, and spontaneous curvature are constant during the process.21 This is because the relative changes of area between final and initial states are relatively large (of order unity). Increase of the area causes a decrease of interfacial concentration, which in turn implies an increase of tension and a decrease of bending constant and (absolute value of) spontaneous curvature. On the other hand, the decrease of interfacial concentration also implies a decrease of the surfactant chemical potential at the interface and should induce a net flux to the interface, which will act to increase back the concentration. The overall time dependence might then be quite complicated. Nevertheless, the growth of the instability does not have to be strongly controlled by the surfactant transport to the interface.21 This is because the instability does not vanish at some critical tension (since C ˜ 0 > 1). Instead, there is a critical spontaneous curvature C ˜ 0 ) 1, below which the system may be stable depending on the surface tension, i.e., if σ- < σ < σ+. Therefore, transport should dominate the evolution only if initially C ˜0 - 1 , 1. In this paper we shall disregard this time dependence, since, as explained above, surfactant conservation effects are not expected to be dramatic in most situations. We shall therefore consider the equations of motion for the cylindrical interface at a constant tension σ˜ ) -1/2 + C ˜ 0. These can be found by solving the Navier-Stokes equations inside and outside the tube with appropriate boundary conditions at the interface,11,17 one of which is the generalized Laplace relation for the (interface) jump in the pressure between the inside and outside.12,22 We consider the high viscous regime (neglect of inertia) because of the mesoscopic length scales involved. Since our equations are limited to the linear regime (of u), the evolution in time is described by a first-order differential equation
dUq ) ω(q)Uq dt
(22)
where
ω(q) ) -
κ Λ(q˜ ) γ(q˜ ) Rc2
(23)
We see that (for C ˜ 0 > 1) q˜ m > 1 and, as shown below, the
Here Λ(q˜ ) accounts for the long range hydrodynamic interaction between different pieces of the interface, which leads to nonlocal dynamics. It plays the role of the wellknown Oseen tensor which describes the hydrodynamic interaction when boundary conditions are taken at infinity. Here, however, one needs to solve the hydrodynamic equations explicitly in order to find Λ. This has been done by Tomotika,17 and we shall simply quote here the result without repeating it. The “hydrodynamic interaction” kernel Λ(q˜ ) was obtained by Tomotika for an arbitrary ratio of oil and water viscosities, and this (complicated) result should be used in the general case. However, since these viscosities usually have roughly the
(18) Dependences of C0 on temperature have been observed by: Strey, R. Colloid Polym. Sci. 1994, 272, 1005. Olsson, U.; Wennerstro¨m, H. Adv. Colloid Interface Sci. 1994, 49, 113. (19) See, for example, T jump in sponge phases: Waton, G.; Porte, G. J. Phys. II 1993, 3, 515.
(20) Note the significant difference with the result of ref 12 for the instability of cylindrical membrane vesicles. In the latter, q˜ m ) 1, independent of the spontaneous curvature. (21) Granek, R.; Ball, R. C.; Cates, M. E. J. Phys. II 1993, 3, 829. (22) Komura, S.; Seki, K. Physica 1993, A192, 27.
Using this in eq 12 we find γ(q) for this case
˜ 0)q˜ 2 + q˜ 4 γ(q) ) 2 - C ˜ 0 - (1 + C
(20)
The tube is therefore unstable if C ˜ 0 > 1. The wavenumber of the energetically most unstable mode qm is given by
q˜ m )
(
)
1+C ˜0 2
1/2
(21)
5026 Langmuir, Vol. 12, No. 21, 1996
Granek
Figure 1. Lowest energy mode (in reduced units) q˜ m ) qmRc and fastest growing mode q˜ * ) q*Rc against the (reduced) spontaneous curvature C ˜ 0 ) C0Rc.
same value, we shall henceforth use, for simplicity, the case of equal viscosities. Then Λ(q˜ ) simplifies to
Λ(q˜ ) )
Φ(q˜ ) )
Φ(q˜ ) 2ηRc
q˜ (I1(q˜ ) K0(q˜ ) - I0(q˜ ) K1(q˜ )) + 2I1(q˜ ) K1(q˜ ) q˜ (I1(q˜ ) K0(q˜ ) + I0(q˜ ) K1(q˜ ))
(24)
(25)
where η is the viscosity. (In(x) and Kn(x) are the modified Bessel functions of order n.) The fastest growing mode q* is found by maximizing ω(q) over q in the band of negative γ(q). The function Φ(q˜ ) has a maximum at q˜ = 1.5 and falls off to zero at q˜ f 0 and at q˜ f ∞. Thus, if q˜ msthe wavenumber that minimizes γ(q)sobeys q˜ m > 1, the same is true for q˜ *, i.e., q˜ * > 1. This is shown in Figure 1, where we plot q˜ * and ˜ 0. We can see that the two wavenumbers are q˜ m against C ˜ 0 . 1, q˜ * becomes almost equal for C ˜ 0 ∼ 1. When C somewhat smaller than q˜ m but remains proportional to it. In fact, from eqs 20 and 23-25 we obtain in this limit q˜ m = (C ˜ 0/2)1/2 and (using asymptotic expansions of the modified Bessel functions for q˜ . 1) q˜ * = (C ˜ 0/3)1/2. Both results scale as C ˜ 01/2, in agreement with Figure 1. Recall that q* gives an estimate for the droplet radius Rs formed by this instability Rs = λ*/4 ) π/(2q*). When C ˜ 0 = 1 we also get q˜ * = 1, and the droplet radius will be close to the tether radius. For C ˜ 0 . 1 the droplet radius is roughly Rs ∼ 1/q* ∼ Rc1/2C0-1/2, namely it is determined by both the new spontaneous curvature and the tether radius. Note that we do not obtain the more intuitive result Rs ∼ C0-1, which is based on energetic considerations alone. Indeed, considering a collection of identical spheres, we find that their energy is minimized, at constant (total) volume and with an interfacial tension σ ) -1/ ˜ 0 . 1), by a radius RsC0 2κC02 (corresponding to the limit C ) 3/2. (We also find, in a similar way to the example given in section 1, that the sphere energy is lower than the cylinder energy if RsC0 > 1.) Thus the fastest growing mode does not necessarily drive the system near to its equilibrium state, although the newly formed droplet state does have a lower energy than the initial cylinder. The growth rate of the fastest growing mode ω(q*) is ˜ 0. plotted in Figure 2 in units of τr-1 ) κ/(ηRc3) against C In the regime 1 < C ˜ 0 < 2, ω(q*) increases sharply from its ˜ 0 . 1, ω(q*) increases vanishing value at C ˜ 0 ) 1. For C
Figure 2. Reduced growth rate of the fastest growing mode ω ˜ (q*) ) ω(q*)ηRc3/κ against the (reduced) spontaneous curvature C ˜ 0 ) C0Rc.
as a power law with increasing C ˜ 0. Indeed, from eqs 2325 we obtain in this limit (or, more precisely, in the limit ˜ 03/2/(6x3). Typical values of swollen q˜ * . 1) ω(q*)τr = C wormlike micellar systems are Rc ∼ 100 Å and κ ∼ 10kBT, and the basic unit time scale is τr ∼ 10-7 s. For a moderate jump where C ˜ 0 = 2, we thus get ω(q*) of order 106 s-1. IV. Conclusions The local structural transformation from cylinder to sphere is thus predicted to occur on a very short time scale of order ∼1 µs.23 This local transformation essentially controls the dynamical phase transition. Of course, further dynamical steps are needed in order to reach the true equilibrium. First, the droplet radius is not quite the equilibrium radius. Such equilibration will require coalescence and breakage of droplets. Second, since surfactant material is conserved, some tubes have to give up their surfactant in favor of others which perform this transformation. The excess swelling fluid (say, the oil) they produce has to be transported to the phase boundaries (“emulsification failure”), and this can be a very slow process. Nevertheless, this transformation can be directly detected, e.g., by looking at the evolution in time of the static structure factor19 (e.g., using small-angle neutron scattering6). Spheres and cylinders have quite different structure factors6,9 so the shape transformation could be clearly detected. The ideal candidate for such experiments would be a system in which the spontaneous curvature is sensitive to a few degrees change of temperature18 (so that one can perform a T jump experiment19). We note the implications on cryotransmission and freeze-fracture electron microscopies on such systems.24 Although these techniques use very fast cooling rates (e.g., 0.1 K/µs), there may be enough time for the cylinder to sphere instability to occur during the cooling process and one may end up with a system of spherical micelles rather than rodlike or wormlike micelles. We also point out the possible relevance of the spontaneous curvature-induced Rayleigh-like instability discussed here to the process of (23) We note again that in this estimate we have neglected the effect of the Gaussian rigidity κj. However, since κj is negative, it only lowers the droplet’s energy so that the breaking rate should not be limited by κj. (24) See, for example: Talmon, Y.; Burns, J. H.; Chestnut, M. H.; Siegel, D. P. J. Electron Microsc. Tech. 1990, 14, 6. Clausen, T. M.; Vinson, P. K.; Minter, J. R.; Davis, H. T.; Talmon, Y.; Miller, W. G. J. Phys. Chem. 1992, 96, 474.
Instability in Swollen Cylindrical Micelles
spontaneous emulsification.4,21 It has been recently suggested that spontaneous curvature can induce buckling of a flat oil-water interface to a state of long cylindricallike fingers.25 It is thus possible that these fingers will break up into spherical droplets following the Rayleighlike instability, which clearly will drive the system closer to its equilibrium state. We conclude with a note of hope that more experimental attention will be given to outof-equilibrium phenomena in self-assembling systems. (25) Hu, J.-G.; Granek, R. J. Phys. II 1996, 6, 999. Granek, R. To be published.
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Acknowledgment. I am grateful to Jian-Guo Hu for stimulating discussions and for verifying some of the equations appearing in the text and to Sam Safran for many useful discussions. This research was supported by the Israel Science Foundation, administered by the Israel Academy of Sciences and Humanities, and by a research grant from the Edward D. and Anna Mitchell Endowment Fund for Excellence in Scientific Achievement. The author is an incumbent of the William T. Hogan and Winifred T. Hogan Career Development Chair. LA9603838
